code: def finite_diff_coeffs(n, approx_order, strategy):
      r"""Generate the finite difference shift values and corresponding
      term coefficients for a given derivative order, approximation accuracy,
      and strategy.
  
      Args:
          n (int): Positive integer specifying the order of the derivative. For example, ``n=1``
              corresponds to the first derivative, ``n=2`` the second derivative, etc.
          approx_order (int): Positive integer referring to the approximation order of the
              returned coefficients, e.g., ``approx_order=1`` corresponds to the
              first-order approximation to the derivative.
          strategy (str): One of ``"forward"``, ``"center"``, or ``"backward"``.
              For the ``"forward"`` strategy, the finite-difference shifts occur at the points
              :math:`x_0, x_0+h, x_0+2h,\dots`, where :math:`h` is some small
              step size. The ``"backwards"`` strategy is similar, but in
              reverse: :math:`x_0, x_0-h, x_0-2h, \dots`. Finally, the
              ``"center"`` strategy results in shifts symmetric around the
              unshifted point: :math:`\dots, x_0-2h, x_0-h, x_0, x_0+h, x_0+2h,\dots`.
  
      Returns:
          array[float]: A ``(2, N)`` array. The first row corresponds to the
          coefficients, and the second row corresponds to the shifts.
  
      **Example**
  
      >>> finite_diff_coeffs(n=1, approx_order=1, strategy="forward")
      array([[-1.,  1.],
             [ 0.,  1.]])
  
      For example, this results in the linear combination:
  
      .. math:: \frac{-y(x_0) + y(x_0 + h)}{h}
  
      where :math:`h` is the finite-difference step size.
  
      More examples:
  
      >>> finite_diff_coeffs(n=1, approx_order=2, strategy="center")
      array([[-0.5,  0.5],
             [-1. ,  1. ]])
      >>> finite_diff_coeffs(n=2, approx_order=2, strategy="center")
      array([[-2.,  1.,  1.],
             [ 0., -1.,  1.]])
  
      **Details**
  
      Consider a function :math:`y(x)`. We wish to approximate the :math:`n`-th
      derivative at point :math:`x_0`, :math:`y^{(n)}(x_0)`, by sampling the function
      at :math:`N<n` distinct points:
  
      .. math:: y^{(n)}(x_0) \approx \sum_{i=1}^N c_i y(x_i)
  
      where :math:`c_i` are coefficients, and :math:`x_i=x_0 + s_i` are the points we sample
      the function at.
  
      Consider the Taylor expansion of :math:`y(x_i)` around the point :math:`x_0`:
  
      .. math::
  
          y^{(n)}(x_0) \approx \sum_{i=1}^N c_i y(x_i)
              &= \sum_{i=1}^N c_i \left[ y(x_0) + y'(x_0)(x_i-x_0) + \frac{1}{2} y''(x_0)(x_i-x_0)^2 + \cdots \right]\\
              & = \sum_{j=0}^m y^{(j)}(x_0) \left[\sum_{i=1}^N \frac{c_i s_i^j}{j!} + \mathcal{O}(s_i^m) \right],
  
      where :math:`s_i = x_i-x_0`. For this approximation to be satisfied, we must therefore have
  
      .. math::
  
          \sum_{i=1}^N s_i^j c_i = \begin{cases} j!, &j=n\\ 0, & j\neq n\end{cases}.
  
      Thus, to determine the coefficients :math:`c_i \in \{c_1, \dots, c_N\}` for particular
      shift values :math:`s_i \in \{s_1, \dots, s_N\}` and derivative order :math:`n`,
      we must solve this linear system of equations.
      """
      if n < 1 or not isinstance(n, int):
          raise ValueError(
              "Derivative order n must be a positive integer."
          )
  
      if approx_order < 1 or not isinstance(approx_order, int):
          raise ValueError(
              "Approximation order must be a positive integer."
          )
  
      num_points = approx_order + 2 * np.floor((n + 1) / 2) - 1
      N = num_points + 1 if n % 2 == 0 else num_points
  
      if strategy == "forward":
          shifts = np.arange(N, dtype=np.float64)
  
      elif strategy == "backward":
          shifts = np.arange(-N + 1, 1, dtype=np.float64)
  
      elif strategy == "center":
          if approx_order % 2 != 0:
              raise ValueError(
                  "Centered finite-difference requires an even order approximation."
              )
  
          N = num_points // 2
          shifts = np.arange(-N, N + 1, dtype=np.float64)
  
      else:
          raise ValueError(
              f"Unknown strategy {strategy}. Must be one of 'forward', 'backward', 'center'."
          )
  
      # solve for the coefficients
      A = shifts ** np.arange(len(shifts)).reshape(-1, 1)
      b = np.zeros_like(shifts)
      b[n] = factorial(n)
  
      # Note: using np.linalg.solve instead of scipy.linalg.solve can cause a bus error when this
      # is inside a tf.py_function inside a tf.function, as occurs with the tensorflow-autograph interface
      # Bus errors were potentially specific to the M1 Mac. Change with caution.
      coeffs = linalg_solve(A, b)
  
      coeffs_and_shifts = np.stack([coeffs, shifts])
  
      # remove all small coefficients and shifts
      coeffs_and_shifts[np.abs(coeffs_and_shifts) < 1e-10] = 0
  
      # remove columns where the coefficients are 0
      coeffs_and_shifts = coeffs_and_shifts[
          :, ~np.all(coeffs_and_shifts == 0, axis=0)
      ]
  
      # sort columns in ascending order according to abs(shift)
      coeffs_and_shifts = coeffs_and_shifts[
          :, np.argsort(np.abs(coeffs_and_shifts)[1])
      ]
      return coeffs_and_shifts
  
file_path: pennylane/pennylane/gradients/finite_difference.py
imports: from typing import Sequence, Callable
  import functools
  from functools import partial
  from warnings import warn
  import numpy
  from scipy.special import factorial
  from scipy.linalg import solve
  import pennylane
  from pennylane.measurements import ProbabilityMP
  from pennylane import transform
  from pennylane.transforms.tape_expand import expand_invalid_trainable
  from pennylane.gradients.gradient_transform import _contract_qjac_with_cjac
  from .general_shift_rules import generate_shifted_tapes
  from .gradient_transform import _all_zero_grad, assert_no_tape_batching, choose_trainable_params, find_and_validate_gradient_methods, _no_trainable_grad
label_desc: Use image node_id 1 for calling a global function with example usage: finite_diff_coeffs(n, approx_order, strategy) and returns: coeffs_and_shifts